If every Dedekind cut is at a rational it seems that these cuts would only produce a countable set and would not produce the whole real line. So how should I think about it.

The point is that there are Dedekind cuts that are NOT at rationals.
For example, take any real number A and consider its decimal expansion. A sequence of rational numbers An can be defined by taking n terms of the expansion. Let the cut be defined by all rationals less than any term in the sequence. This cut gives the real number A.

Even though all Dedekind cuts consist of only rational numbers, all are not rational cuts.
T = { x [itex]\in[/itex] Q: x^2 < 2 or x < 0 } is a dedekind cut, you can check that all the properties hold, but it can not be a rational cut.